Infinitesimal Newton-Okounkov bodies on products of curves

TL;DR

The paper determines the generic infinitesimal Newton-Okounkov body $\Delta_x(L)$ for box-product polarizations on products of curves, proving that in any dimension $n$ it is the simplex with explicit vertices $$(0,\dots,0), (1,1,0,\dots,0), (2,0,2,0,\dots,0), \dots, (n-1,0,\dots,0,n-1), (n,0,\dots,0).$$ It achieves this by analyzing vertical slices $\Delta_x(L)_{\nu_1=t}$ via a graded linear series $W^t_{\bullet}$ and identifying these with complete linear series on a toric modification $X_{\Sigma}$, allowing convex-geometric control of asymptotics. The paper provides explicit descriptions in small dimensions ($n=2,3$) with concrete vertex sets, and proves a general existence result for balanced box-products in arbitrary dimension, where the iNObody is exactly the stated simplex. These results give the first nontrivial description of generic iNObodies in arbitrary dimension and connect local positivity invariants to toric geometry and convex bodies.

Abstract

We compute the generic infinitesimal Newton-Okounkov body at any point for some box-product polarizations on products of curves. This appears to be the first nontrivial description of such a body in arbitrary dimension.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1: Positive cones on $X$ and $\overline X$
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• proof : Proof of Theorem \ref{['thm:smalldimensions']}